This from 2008 isn't great, but it's not calling it woo: http://www.nhs.uk/news/2008/10October/Pages/Honeyandburns.as...
This from 2016 is a bit better: http://www.cochrane.org/CD005083/WOUNDS_honey-as-a-topical-t...
> There is high quality evidence that honey heals partial thickness burns around 4 to 5 days more quickly than conventional dressings. There is moderate quality evidence that honey is more effective than antiseptic followed by gauze for healing wounds infected after surgical operations.
I think "partial thickness" is called "second degree" in the US.
That's different from just pouring it on a new burn, and I definitely don't think people should do that.
> Having had oil burns heal without honey
I'd say you need to have Terminator-like skin in order to (relatively) quickly heal your skin (with no scars) after pouring more than half of pan of burning oil on your arm.
http://www.academia.edu/2189571/Pdf_6_The_antibacterial_acti...